The fundamental problem in munition guidance is determining the flight path that a missile or other guided munition needs to take to in order to reach a given target. More specifically, the problem is determining the controls that should be applied to the munition during its flight in order to reach the target, for example determining the lateral acceleration that is needed to achieve the desired flight path.
Classical approaches to guidance involve selecting a geometric law governing the flight path, for example line-of-sight guidance, pure pursuit or parallel navigation, and then determining and applying to the munition the control that is needed, at any given moment during its flight, to minimise the error between the actual state (e.g. position, velocity) of the munition and the ideal state determined by the guidance law. For the case of parallel navigation, the most commonly used corresponding guidance law is proportional navigation (PN). Parallel navigation is the constant-bearing rule, so that the direction of the line-of-sight from the munition to the target is kept constant relative to an inertial frame of reference, i.e. in the planar case, {dot over (λ)}=0 where λ is the angle between the munition-target line of sight and a reference line in the inertial frame; the PN guidance law is then aMc=k{dot over (λ)}, where aMc is the command for lateral acceleration and k is a constant (gain); thus, the lateral acceleration applied to the munition is proportional to the rate of change with time of angle λ.
Many modern guidance systems rely on an alternative class of techniques, known collectively as optimal guidance. In optimal guidance, rather than determining the control in view of a geometric law, the control is determined with a view to minimising a cost function. The cost function is typically a function of the miss distance, i.e. the distance by which the munition will miss its target, and the control effort, i.e. the amount of effort required to reach the target (e.g. the amount of fuel required). The guidance problem typically also involves constraints, for example relating to the dynamics of the munition. Solution of the optimisation problem, analytically or by computer-implemented numerical methods, provides the optimal control, which minimises the cost function.
In practical guidance situations, a target state is usually not known and has to be determined through measurements. The measurements are used to determine an evolving probability distribution of the future location of the target, to which the pursuer is to be optimally guided. When the location of the target can be described using a Gaussian probability distribution, guidance is relatively straightforward. However, when the target distribution cannot be described using a Gaussian probability distribution, for example when the target location is limited to a number of possible discrete locations (e.g. a number of possible road routes that the target may be taking), the guidance problem becomes more difficult.
The classic way to represent the target probability distribution when it is non-Gaussian is to use a Particle Filter (introduced by N. J. Gordon, D. J. Salmond, and A. F. M. Smith, in their seminal paper Novel approach to nonlinear/non-Gaussian Baysian state estimation, IEE Proc.—F, vol 140, No 2, pp 107-113, 1993). M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp (in A tutorial on particle filters for non-linear/non-gaussian Bayesian tracking, IEE Trans. Signal processing, Vol. 50, February 2002) provide an introductory tutorial on Particle Filtering, and A. Doucet, N. de Freitas, and N. Gordon (editors, in Sequential Monte Carlo Methods in Practice. Springer-Verlang, 2001) give a collection of papers describing the theory behind Particle Filtering, together with various improvements to and applications of Particle Filtering. Particle Filtering essentially describes the future target distribution using a weighted sample of target trajectories from the present target distribution. It simplifies the implementation of the Bayesian rule for measurement updates by modifying just the particle weights and then resampling after each measurement. The resulting weighted target trajectory population is used to calculate approximately an expected value of a function of the target state as a random variable. That is done just by applying the function to the target trajectory sample and then forming a weighted sum of the results using the particle filter weights. Although that provides a way to measure and predict target properties, it does not deal with how to guide the pursuer to the target.
One of the outstanding problems in Guidance and Control is to guide a pursuer to a target that has genuinely non-Gaussian statistics, so that the guidance algorithm gives the best average outcome against miss distance, pursuer manoeuvring and pursuer state costs, subject to pursuer state constraints. D. J. Salmond, N. O. Everett, and N J Gordon (“Target tracking and guidance using particles”, in Proceedings of the American Control Conference, pages 4387-4392, Arlington, Va., Jun. 25-27, 2001) pioneered this area by using a simple guidance law, with a fixed shape, and in a following paper (D Salmond and N. Gordon, “Particles and mixtures for tracking and guidance”, in A. Doucet, N. de Freitas, and N. Gordon, editors, Sequential Monte Carlo Methods in Practice, pages 518-532. Springer-Verlang, 2001) classified several types of suboptimal guidance solutions.
It would be advantageous to provide an improved method and apparatus for guidance to a target having non-Gaussian statistics.